3 Description

Balancing may reduce the 1-norm of the matrices and improve the accuracy of the computed eigenvalues and eigenvectors in the complex generalized eigenvalue problem

Ax=λBx.

F08WVF (ZGGBAL) is usually the first step in the solution of the above generalized eigenvalue problem. Balancing is optional but it is highly recommended.

The term ‘balancing’ covers two steps, each of which involves similarity transformations on A and B. The routine can perform either or both of these steps. Both steps are optional.

The routine first attempts to permute A and B to block upper triangular form by a similarity transformation:

PAPT=F=F11F12F13F22F23F33

PBPT=G=G11G12G13G22G23G33

where P is a permutation matrix, F11, F33, G11 and G33 are upper triangular. Then the diagonal elements of the matrix pairs F11,G11 and F33,G33 are generalized eigenvalues of A,B. The rest of the generalized eigenvalues are given by the matrix pair F22,G22 which are in rows and columns ilo to ihi. Subsequent operations to compute the generalized eigenvalues of A,B need only be applied to the matrix pair F22,G22; this can save a significant amount of work if ilo>1 and ihi<n. If no suitable permutation exists (as is often the case), the routine sets ilo=1 and ihi=n.

The routine applies a diagonal similarity transformation to F,G, to make the rows and columns of F22,G22 as close in norm as possible:

DFD^=I000D22000IF11F12F13F22F23F33I000D^22000I

DGD-1=I000D22000IG11G12G13G22G23G33I000D^22000I

This transformation usually improves the accuracy of computed generalized eigenvalues and eigenvectors.

On exit: details of the permutations and scaling factors applied to the left side of the matrices A and B. If Pi is the index of the row interchanged with row i and di is the scaling factor applied to row i, then

LSCALEi=Pi, for i=1,2,…,ilo-1;

LSCALEi=di, for i=ilo,…,ihi;

LSCALEi=Pi, for i=ihi+1,…,n.

The order in which the interchanges are made is n to ihi+1, then 1 to ilo-1.

6 Error Indicators and Warnings

If INFO=-i, argument i had an illegal value. An explanatory message is output, and execution of the program is terminated.

7 Accuracy

The errors are negligible, compared to those in subsequent computations.

8 Further Comments

F08WVF (ZGGBAL) is usually the first step in computing the complex generalized eigenvalue problem but it is an optional step. The matrix B is reduced to the triangular form using the QR factorization routine F08ASF (ZGEQRF) and the unitary transformation Q is applied to the matrix A by calling F08AUF (ZUNMQR). This is followed by F08WSF (ZGGHRD) which reduces the matrix pair into the generalized Hessenberg form.

If the matrix pair A,B is balanced by this routine, then any generalized eigenvectors computed subsequently are eigenvectors of the balanced matrix pair. In that case, to compute the generalized eigenvectors of the original matrix, F08WWF (ZGGBAK) must be called.

The total number of floating point operations is approximately proportional to n2.